Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=x$$

Step-by-Step Solution

Answer

The solutions to the equation \(y = x\) are \((0,0)\), \((1,1)\), \((2,2)\), \((-1,-1)\), \((-2,-2)\). These points, when plotted on a graph, form a straight line.

Step 1: Identify the Linear Equation

The given linear equation in this problem is \(y = x\). This is a simple linear equation where the coefficient of \(x\) is 1 and there is no constant term.

Step 2: Generate the Solutions

Since \(y = x\), solutions to the equation can be found by selecting any number for \(x\) and then setting \(y\) to the same value. Here are five simple solutions: \((0,0)\), \((1,1)\), \((2,2)\), \((-1,-1)\), \((-2,-2)\)

Step 3: Plot the Values on a Graph

Place dots on the coordinates of each of these five solutions on a graph with \(x\) as the horizontal axis and \(y\) as the vertical axis. By connecting these dots, a straight line is formed. This line represents the linear equation \(y = x\) and every point on this line is a solution to the equation.

Key Concepts

Understanding Linear Equation Solutions

In the world of algebra, linear equations are the backbone of coordinate graphing. A linear equation like the one in our example, y = x, depicts a relationship between two variables, where the value of one variable depends on the value of the other. Solutions to linear equations are essentially pairs of numbers that make the equation true.

For the equation y = x, any pair where the x and y values are equal will be a solution because it aligns with the initial condition set by the equation. The solutions are infinite but selecting a few specific ones helps us to graph the equation and visualize the relationship between x and y. A crucial aspect of algebra is not just finding the solutions but also understanding their implications on graphing and coordinate relationships.

Coordinate Graphing Essentials

Coordinate graphing is a method of representing algebraic equations visually. The coordinate plane, otherwise known as the Cartesian plane, has two axes: horizontal (x-axis) and vertical (y-axis). Each point on the graph corresponds to a pair of (x, y) values, which represent the coordinates.

Take our linear equation y = x. When we plot the solutions on a graph, what we're actually doing is translating numerical information into visual format. The points from our example, such as (0,0) or (2,2), will appear on a straight line that bisects the plane at a 45-degree angle. This line is a graphical representation of all the possible solutions to the equation, demonstrating the one-to-one relationship between the x and y values.

The beauty of coordinate graphing lies in its ability to show us at a glance how two variables relate across a continuum of values. This helps in predicting and interpreting the nature of the linear relationship.

The Role of a Table of Values

A table of values is a structured method to organize and list pairs of numbers (coordinates) that fulfill an algebraic equation. It serves as a foundational step in the graphing process.

For the equation y = x, a table would provide a clear and concise representation of its solutions. To construct this table, one simply chooses a series of x values, then applies the equation y = x to find the corresponding y values. The list of ordered pairs (x, y) becomes the set of points that you will graph. For instance, your table for our equation might start like this:

• (0, 0)
• (1, 1)
• (2, 2)
• (-1, -1)
• (-2, -2)

Tables of values are indispensable tools in mathematics because they organize information that aids in recognizing patterns, forming predictions, and graphically depicting equations. They enable us to move from abstract equations to concrete visual representations smoothly.